How would someone get these raised speckle dots on a glazed piece?

What you're describing sounds like glaze dotting done over a fired glaze, then refired. It's a real technique and pretty forgiving once you know a few things. The way it usually works: you make or buy a glaze that's a different color from your base, fire the piece to full temp with just the base glaze, then come back and dab dots of the second glaze straight onto the fired surface and refire the whole thing. The dots melt slightly and round up into little beads, exactly like you're seeing on your aunt's dish. If the dots are applied thick, almost like toothpaste, they don't flatten out much and you get that raised texture. The thin dark eye outlines sound like a different approach, probably an underglaze or oxide wash done before the main glaze firing, which is why they sit flat while the spots sit proud.

The other possibility is the majolica method, where you paint colored glazes on top of an unfired base glaze and fire everything once. On majolica pieces the colors do seem to float on the surface. But if the spots catch the light like actual beads rather than flat pools of color, I'd put my money on the dot-and-refire route.

For your planter at cone 6, the important bit is that the dot glaze and your base glaze should be reasonably compatible, meaning similar expansion. If they're way off you can get crazing around each dot or the dots can flake off the edges later. If you're mixing your own, stick with glazes from the same recipe family. If you're buying commercial, use the same brand's line and you're usually fine.

Practical tips: scuff or clean the fired glaze surface before dotting so the new glaze bonds well, apply dots with a squeeze bottle or a brush for even beads, and refire at a temp where that dot glaze will actually melt. If a test dot comes out flat instead of beaded, your glaze is too runny and needs a touch more clay or silica. Do a couple test tiles before committing to the planter. One afternoon of testing saves you a lot of heartache on the real piece.

That shimmery purple makes me think of the old china-painting enamel work my mother did back in the day. Ladies fired colored enamels on top of allready-glazed ware at a low temp, just enough to melt the dots, and they stayed proud and glassy like you're describing. Worth checking wether the dots have hard defined edges or melted-soft ones. Either way, for your planter you can get that beaded look by firing the dots just to a gloss, not to a full melt. An old trick for even dots: the butt end of a paintbrush dipped and touched down, works better than any squeeze bottle I ever used.

Agreed with the dot-and-refire take here. I've done this on a few mugs at cone 6 and the bead effect is real when you load the dots on thick. One thing I learned the hard way: let the dots dry completley before you move the piece or even brush past it. My first batch smudged from a knuckle grazing one spot, and that dot smeared into a teardrop shape in the refire instead of rounding up. Also hold the piece under a lamp at an angle before firing, because a dot that looks covered can actually be sitting thin in the middle and will go flat on you.